Properties

Degree 2
Conductor $ 11 \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 3·5-s + 2·7-s − 2·9-s + 11-s − 2·12-s + 2·13-s + 3·15-s + 4·16-s − 17-s + 2·19-s − 6·20-s + 2·21-s − 3·23-s + 4·25-s − 5·27-s − 4·28-s − 6·29-s − 7·31-s + 33-s + 6·35-s + 4·36-s − 7·37-s + 2·39-s + 12·41-s − 10·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1.34·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 0.774·15-s + 16-s − 0.242·17-s + 0.458·19-s − 1.34·20-s + 0.436·21-s − 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.755·28-s − 1.11·29-s − 1.25·31-s + 0.174·33-s + 1.01·35-s + 2/3·36-s − 1.15·37-s + 0.320·39-s + 1.87·41-s − 1.52·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{187} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 187,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.409757951$
$L(\frac12)$  $\approx$  $1.409757951$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 - T \)
17 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.71890257312052, −18.59243579166473, −17.87325923028080, −17.41375207116548, −16.45308721021524, −14.84825696286607, −14.30186090895599, −13.66152403485290, −12.99782247857233, −11.58530592644471, −10.41422832511990, −9.312607292057117, −8.847231817862949, −7.753924260656871, −5.987854794252313, −5.176237652424836, −3.642155695833220, −1.876642088477970, 1.876642088477970, 3.642155695833220, 5.176237652424836, 5.987854794252313, 7.753924260656871, 8.847231817862949, 9.312607292057117, 10.41422832511990, 11.58530592644471, 12.99782247857233, 13.66152403485290, 14.30186090895599, 14.84825696286607, 16.45308721021524, 17.41375207116548, 17.87325923028080, 18.59243579166473, 19.71890257312052

Graph of the $Z$-function along the critical line